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Abstract:
Let K = \bbfQ(\sqrt-d) be an imaginary quadratic field, d gt; 3, d \equiv 3 \pmod 4, and let \psi be a grössencharacter in K satisfying the condition: \psi(α) = \varepsilon(α)α\sp2k-1 for α \in K\sp*, where \varepsilon is a quadratic character of conductor (\sqrt-d) and \varepsilon(n) = (n\over d) for n\in \bbfZ. The authors give an explicit formula for L(\psi,k), in terms of the special values of the k-th non-holomorphic derivative of the θ-series θ\sb1/2(z) = \sum\sp ∞\sbℓ = 0 \textexp(π\sb i(2ℓ+1)\sp 2 z/4) at z = z\sb j, z\sb j \in K, 1 ≤q j ≤q h, where h is the class number of K. As a consequence of their formula, they note that L(\psi,k) ≥q 0, as has been conjectured by \it R. Greenberg [Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)].\par Further, for K = \bbfQ(\sqrt-7) the authors prove that L(\psi,k) = R\sb kB(k)\sp 2 with B(k) \in \bbfZ, R\sb k being the (explicitly given) ``transcendental part'' of L(\psi,k), as has been previously conjectured by \it B. H. Gross and \it D. Zagier [Mém. Soc. Math. Fr. 108, 49-54 (1980; Zbl 0462.14015)].
